Simplify the following expression: $\dfrac{40r^3}{24r^5}$ You can assume $r \neq 0$.
Solution: $ \dfrac{40r^3}{24r^5} = \dfrac{40}{24} \cdot \dfrac{r^3}{r^5} $ To simplify $\frac{40}{24}$ , find the greatest common factor (GCD) of $40$ and $24$ $40 = 2 \cdot 2 \cdot 2 \cdot 5$ $24 = 2 \cdot 2 \cdot 2 \cdot 3$ $ \mbox{GCD}(40, 24) = 2 \cdot 2 \cdot 2 = 8 $ $ \dfrac{40}{24} \cdot \dfrac{r^3}{r^5} = \dfrac{8 \cdot 5}{8 \cdot 3} \cdot \dfrac{r^3}{r^5} $ $\phantom{ \dfrac{40}{24} \cdot \dfrac{3}{5}} = \dfrac{5}{3} \cdot \dfrac{r^3}{r^5} $ $ \dfrac{r^3}{r^5} = \dfrac{r \cdot r \cdot r}{r \cdot r \cdot r \cdot r \cdot r} = \dfrac{1}{r^2} $ $ \dfrac{5}{3} \cdot \dfrac{1}{r^2} = \dfrac{5}{3r^2} $